The bad news is that I need to solve for the initial height, [itex]h[/itex] in terms of all the other variables, and there seems to be no easy way to make things work. I tried assigning the term [itex]r = H-h[/itex], but it's still a huge pain to solve

The equation I get is [tex]\frac {a} {\sqrt {b - z}} - \sqrt {z} - 1 = 0[/tex] where [tex]a = \frac {d} {2H}[/tex][tex]b = \frac {v^2} {2gH}[/tex][tex]z = 1 - \frac h H[/tex] This can be solved numerically. Or, after some more massage, it could be converted into a quartic equation that could be solved by Ferrari's method, or numerically. The latter is bit tricky, because the quartic has four roots, but there is only one physical solution. The physical solution has these properties: [itex] z \ge 0 [/itex] (because [itex]h \le H[/itex]) and [itex]z \lt b [/itex] (because initial kinetic energy must be greater than required for purely vertical motion from [itex]h[/itex] to [itex]H[/itex]). Note also that not all combinations of parameters admit a solution.